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Continuity of Discrete Homomorphisms of Diffeomorphism Groups

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Continuity of Discrete Homomorphisms of Diffeomorphism Groups Geometry & Topology 19 (2015) 2117–2154 msp Continuity of discrete homomorphisms of diffeomorphism groups SEBASTIAN HURTADO Let M and N be two closed C 1 manifolds and let Diffc.M / denote the group of C 1 diffeomorphisms isotopic to the identity. We prove that any (discrete) group homomorphism between Diffc.M / and Diffc.N / is continuous. We also show that a nontrivial group homomorphism ˆ Diffc.M / Diffc.N / implies that dim.M / W ! Ä dim.N / and give a classification of such homomorphisms when dim.M / dim.N /. D 57M99; 55Q33, 55Q32 1 Introduction Let M be a C 1 manifold and let Diffc.M / be the group of C 1 diffeomorphisms of M which are isotopic to the identity and compactly supported (if M is noncompact). A well-known theorem of Mather, Herman, Thurston and Epstein states that Diffc.M / as a discrete group is simple (see Banyaga[4]). In[7], Filipckewicz proved the following result: Given two C 1 manifolds M and N , the groups Diffc.M / and Diffc.N / are isomorphic (as discrete groups) if and only if the manifolds M and N are C 1 diffeomorphic. Moreover, if such an isomorphism exists it is given by conjugation with a fixed element of Diff.M /. Filipckewicz’s theorem implies in particular that two different smooth exotic 7–spheres have different diffeomorphism groups. The focus of this paper is the study of abstract group homomorphisms between groups of diffeomorphisms. We will make no assumptions on the continuity of the homo- morphisms. Let M and N be two manifolds and let ˆ Diffc.M / Diffc.N / be a W ! group homomorphism. Observe that the simplicity of these groups implies that ˆ is either trivial or injective. Some examples of homomorphisms of the type ˆ Diffc.M / Diffc.N / are the W ! following. (1) Inclusion If M N is an open submanifold of N , the inclusion gives a homomorphism ˆ Diffc.M / Diffc.N /. W ! Published: 29 July 2015 DOI: 10.2140/gt.2015.19.2117 2118 Sebastian Hurtado (2) Coverings If p N M is a finite covering, every diffeomorphism f W ! 2 Diffc.M / can be lifted to a diffeomorphism fz of N . These lifts in many cases give a group homomorphism. For example, this is the case when p is an irregular cover (ie Deck.M=N / Id ), as in this case he lift f of any diffeomorphism f D f g z is unique. (3) Bundles The unit tangent bundle UT .M /, the Grassmannian Grk .M /, con- sisting of k –planes in T .M /, product bundles, symmetric products and other bundles over M admit a natural action of Diffc.M /. Observe that all the homomorphisms described above are in some sense continuous maps from Diffc.M / to Diffc.N / in their usual C 1 topologies. The main result of this paper shows that this is always the case when M is closed. Before stating our main result we need the following definition: Definition 1.1 For any compact subset K M , let DiffK .M / denote the group of dif-  feomorphisms in Diffc.M / supported in K. A group homomorphism ˆ Diffc.M / W ! Diffc.N / is weakly continuous if for every compact set K M , the restriction  ˆ of ˆ to DiffK .M / is continuous in the weak topology; see Section 2.3. jDiffK .M / Our main theorem is the following: Theorem 1.2 Let M and N be C manifolds. If ˆ Diffc.M / Diffc.N / is a 1 W ! discrete group homomorphism, then ˆ is weakly continuous. It is worth pointing out that there is no assumption on the dimensions of the manifolds. As a consequence of Theorem 1.2, we will obtain a classification of all possible homomorphisms in the case when dim.M / dim.N /. Mann[16] gave a classification of all the nontrivial homomorphisms ˆ Diffc.M / W ! Diffc.N / in the case that N is 1–dimensional. She showed using one-dimensional dynamics techniques (Kopell’s lemma, Szekeres theorem etc) that M must be one- dimensional and that these homomorphisms are what she described as “topologically diagonal”. In the case when M N R, topologically diagonal means that ˆ is given in D D the following way: take a collection of finite disjoint open intervals Ij R and diffeomorphisms fj M Ij . For each diffeomorphism g Diffc.R/ the action W ! 2 1 of ˆ.g/ in the interval Ij is given by conjugation by fj , ie ˆ.g/.x/ figf .x/ D i for every x Ii . The action is defined to be the identity everywhere else. 2 We will prove a generalization of the previous result: Geometry & Topology, Volume 19 (2015) Continuity of homomorphisms of diffeomorphism groups 2119 Theorem 1.3 Let M and N be C manifolds, let N be closed and dim.M / 1 dim.N /. If ˆ Diffc.M / Diffc.N / is a nontrivial homomorphism of groups, then W ! dim.M / dim.N / and ˆ is “extended topologically diagonal”. D The definition of extended topologically diagonal is given in Definition 3.12 and is a generalization of what we previously described as topologically diagonal by allowing the possibility of taking finite coverings of M and embedding them into N ; see Definition 3.12. Observe that as a corollary of Theorem 1.3 we obtain another proof of Filipckewicz’s theorem. The result also gives an affirmative answer to a question of Ghys (see[11]) asking whether a nontrivial homomorphism ˆ Diffc.M / Diffc.N / implies that W ! dim.N / dim.M /. The rigidity results we obtain can be compared to analogous results obtained by Aramayona and Souto[1], who proved similar statements for homomorphisms between mapping class groups of surfaces and to deep results of Margulis about rigidity of lattices in linear groups; see[21, Chapter 5]. 1.1 Ingredients and main idea of the proof The key ingredient in the proof of Theorem 1.2 is a theorem of E Militon. This theorem says roughly that given a sequence hn converging to the identity in Diffc.M / suffi- ciently fast (a geometric statement), one can construct a finite subset S of Diffc.M /, such that the group generated by S contains the sequence hn and such that the word length lS .hn/ (see Definition 2.2) of each diffeomorphism hn in the alphabet S is bounded by a sequence kn not depending on hn (an algebraic statement); see Theorem 2.3 for the correct statement. Militon’s theorem is a generalization of a result of Avila[3] about Diff .S1/ and is related to results of Calegari and Freedman[6]. Militon’s result is strongly related to the theme of distortion elements in geometry group theory; see Gromov[12, Chapter 3]. An element f of a finitely generated group G n with generating set S is called distorted if limn lS .f /=n 0. !1 D Consider for example the group BS.2; 1/ a; b bab 1 a2 . This group can be D f j D g thought as the group generated by the functions a x x 1 and b x 2x in n W 7! C W 7! Diff.R/. Observe that bnab n a2 for every n and therefore a is distorted (in some D sense a is “exponentially” distorted). An element f Diffc.M / is said to be recurrent in Diffc.M / if f satisfies that n2 1 lim inf dC .f ; Id/ 0, for example an irrational rotation in S is recurrent (see 1 D Section 2.3). A corollary of Militon’s theorem is that any recurrent element in Diffc.M / Geometry & Topology, Volume 19 (2015) 2120 Sebastian Hurtado is distorted in some finitely generated subgroup of Diffc.M /. Even more, it implies that such an element is “arbitrarily distorted” in the sense that f could be made as distorted as one wants (see[6] for a precise definition). For a nice exposition of the concept of distortion in transformation groups and the definition of the different types of distortion, see[6]. Observe that distorted elements are preserved under group homomorphisms and there- fore if one wants to understand the existence of a group homomorphism, it might be fruitful to understand the distortion elements of the groups involved in the homomor- phism. One example of this approach can be seen in the work of Franks and Handel[10]. They proved that any homomorphism from a large class of higher rank lattices (for example SL3.Z/) into Diff.S/ (the group of diffeomorphisms of a surface preserving a measure ) is finite using the fact that has distorted elements and showing that any distorted element f Diff.S/ fixes the support of the measure . For a more 2 concrete statement of this and a clear exposition, see Franks[9]. The main idea of how to use Militon’s theorem to obtain our rigidity results is illustrated in Section 2.2 with a motivating example that encloses the main idea of Theorem 1.2 and is the heart of this paper. 1.2 Outline The paper is divided as follows. In Section 2, we will use Militon’s theorem to prove Lemma 2.1, which is the main tool to prove Theorem 1.2. In Section 3, we prove Theorem 1.3 assuming that ˆ is weakly continuous; this section is independent of the other sections. In Section 4, we establish some general facts about two constructions in group actions on manifolds that we will use for the proof of Theorem 1.2. In Section 5, we show that ˆ is weakly continuous using the results of Section 2 and the facts discussed in Section 4, finishing the proof of Theorems 1.2 and 1.3.
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